## View abstract

### Session S30 - Mathematical Methods in Quantum Mechanics

Friday, July 16, 19:30 ~ 19:55 UTC-3

## Semiclassical resolvent bound for long range Lipschitz potentials

### Jacob Shapiro

#### University of Dayton, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc4033cd6076c7ce05e44884c8db533c5').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc4033cd6076c7ce05e44884c8db533c5 = 'jsh&#97;p&#105;r&#111;1' + '&#64;'; addyc4033cd6076c7ce05e44884c8db533c5 = addyc4033cd6076c7ce05e44884c8db533c5 + '&#117;d&#97;yt&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_textc4033cd6076c7ce05e44884c8db533c5 = 'jsh&#97;p&#105;r&#111;1' + '&#64;' + '&#117;d&#97;yt&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakc4033cd6076c7ce05e44884c8db533c5').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc4033cd6076c7ce05e44884c8db533c5 + '\'>'+addy_textc4033cd6076c7ce05e44884c8db533c5+'<\/a>';

We present an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2\Delta + V(x) -E$ in dimension $n\neq 2$, where $h, E >0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit long range decay at infinity, and may grow like a sufficiently small negative power of $r$ as $r\to\infty$. The resolvent norm grows exponentially in $h^{ −1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{−1/2}$ for some $C > 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.

Joint work with Jeffrey Galkowski (University College London).